On a motivic invariant of the arc-analytic equivalence

Campesato, Jean-Baptiste

Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 143-196.

À un germe Nash, nous associons une fonction zêta similaire à la fonction zêta motivique de J. Denef et F. Loeser. Il s’agit d’une série formelle à coefficients dans un anneau de Grothendieck $ℳ$ des ensembles $\mathrm{𝒜𝒮}$ au-dessus de ${ℝ}^{*}$ à $\mathrm{𝒜𝒮}$-bijection ${ℝ}^{*}$-équivariante près. Cet anneau de Grothendieck est analogue à celui construit par G. Guibert, F. Loeser et M. Merle. Cette fonction zêta généralise les précédentes constructions de G. Fichou. Sa richesse algébrique permet d’obtenir une formule de convolution ainsi qu’une formule de type Thom–Sebastiani.

On démontre que la fonction zêta considérée dans cet article est un invariant de l’équivalence arc-analytique, une caractérisation de l’équivalence blow-Nash de G. Fichou. La formule de convolution permet d’obtenir une classification partielle des polynômes de Brieskorn à équivalence arc-analytique près. Plus précisément, on montre que le type arc-analytique d’un tel polynôme détermine ses exposants.

To a Nash function germ, we associate a zeta function similar to the one introduced by J. Denef and F. Loeser. Our zeta function is a formal power series with coefficients in the Grothendieck ring $ℳ$ of $\mathrm{𝒜𝒮}$-sets up to ${ℝ}^{*}$-equivariant $\mathrm{𝒜𝒮}$-bijections over ${ℝ}^{*}$, an analog of the Grothendieck ring constructed by G. Guibert, F. Loeser and M. Merle. This zeta function generalizes the previous construction of G. Fichou but thanks to its richer structure it allows us to get a convolution formula and a Thom–Sebastiani type formula.

We show that our zeta function is an invariant of the arc-analytic equivalence, a version of the blow-Nash equivalence of G. Fichou. The convolution formula allows us to obtain a partial classification of Brieskorn polynomials up to arc-analytic equivalence by showing that the exponents are arc-analytic invariants.

Classification : 14P20, 14E18, 14B05
Keywords: real singularities, Nash functions, motivic integration, arc-analytic functions, blow-Nash equivalence, arc-analytic equivalence
Mot clés : singularités réelles, fonctions Nash, intégration motivique, fonctions analytiques par arcs, équivalence blow-Nash, équivalence arc-analytique

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